Provability interpretations of modal logic

@article{Solovay1976ProvabilityIO,
  title={Provability interpretations of modal logic},
  author={R. Solovay},
  journal={Israel Journal of Mathematics},
  year={1976},
  volume={25},
  pages={287-304}
}
  • R. Solovay
  • Published 1976
  • Mathematics
  • Israel Journal of Mathematics
We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev* ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and let (□ψ)* be a formalization of “ψ)* is a theorem ofP”. We say that a modal formula, χ, isvalid if ψ* is a theorem ofP in each such interpretation. We provide an axiomitization of the class of valid formulae and prove that this class is recursive. 
View on Springer
404 Citations
Highly Influential Citations
22
Background Citations
80
Methods Citations
30
Results Citations
7

404 Citations

Citation Type

Citation Type

The Logic of Arithmetical Hierarchy
TLDR
An axiomatization and decidability of the sets of the modal formulas which are schemata of: (1) PA-provable, (2) true arithmetical sentences. Expand
Provability, truth, and modal logic
  • G. Boolos
  • Mathematics, Computer Science
  • J. Philos. Log.
  • 1980
TLDR
Certain systems of (propositional) modal logic that are of special interest in connection with the study of the notions of provability in P(eano) A(arithmetic) and truth in the standard model for PA are discussed. Expand
On the Degrees of Unsolvability of Modal Predicate Logics of Provability
  • Vann McGee
  • Mathematics, Computer Science
  • J. Symb. Log.
  • 1994
TLDR
Vardanyan [11] shows that, provided that MPL(A) will be the set of modal formulas q such that, for every interpretation *, q$* is a theorem of A. Expand
Provability as a Modal Operator with the models of PA as the Worlds
This paper introduces a propositional modal model of Gödel-Löb Logic whose worlds are the models of Peano Arithemetic and whose box modality is equivalent to an operator satisfying the Hilbert… Expand
Sequent logic of arithmetic decidability ∗
Our paper continues, on the one hand, the study of modal logics that have arithmetical semantics, and on the other, the investigation of decidability (or “non-contingency”) logics. We present… Expand
Finite Kripke Models and Predicate Logics of Provability
The paper proves a predicate version of Solovay's well-known theorem on provability interpretations of modal logic: If a closed modal predicate-logical formula R is not valid in some finite Kripke… Expand
On Strong Provability Predicates and the Associated Modal Logics
TLDR
Some further modal properties of Dzhaparidze's logic are proved, e.g., the fixed point property and the Craig interpolation lemma and other examples of the strong provability predicates and their applications. Expand
Predicate provability logic with non-modalized quantifiers
TLDR
It is proved the recursive enumerability of the sets of Q′-formulas all arithmetical examples of which are: 1) T-provable, 2) true. Expand
Logical equations and admissible rules of inference with parameters in modal provability logics
  • V. Rybakov
  • Mathematics, Computer Science
  • Stud Logica
  • 1990
TLDR
It is proved that the decision problem for admissibility of rules in Gödel-Löb systemGL and Solovay logicS is decidable and the problem of the existence of finite basis for admissible rules forGL andS has a negative solution. Expand
Bimodal Logics for Extensions of Arithmetical Theories
TLDR
The bimodal provability logics for certain natural pairs of recursively enumerable theories, mostly related to fragments of arithmetic, are characterized and modal logics with propositional constants with natural Kripke semantics are introduced. Expand
...
1
2
3
4
5
...

References

SHOWING 1-4 OF 4 REFERENCES
Semantical Analysis of Modal Logic I Normal Modal Propositional Calculi
Publisher Summary This chapter discusses semantical analysis of modal logic ii and non-normal modal propositional calculi. The proof of sufficiency, which is omitted by many, proceeds by constructing… Expand
Solution of a Problem of Leon Henkin
  • M. H. Lob
  • Mathematics, Computer Science
  • J. Symb. Log.
  • 1955
TLDR
This note presents a solution of the previous problem with respect to the system Z μ, and, more generally, to any system whose set of theorems is closed under the rules of inference of the first order predicate calculus, and satisfies the subsequent five conditions, and in which the function ( k, l ) used below is definable. Expand
One Hundred and Two Problems in Mathematical Logic
  • H. Friedman
  • Mathematics, Computer Science
  • J. Symb. Log.
  • 1975
This expository paper contains a list of 102 problems which, at the time of publication, are unsolved. These problems are distributed in four subdivisions of logic: model theory, proof theory and… Expand
Introduction to Metamathematics